People reason about real-estate prices both in terms of general rules and in terms of analogies to similar cases. We propose to empirically test which mode of reasoning fits the data better. To this end, we develop the statistical techniques required for the estimation of the case-based model. It is hypothesized that case-based reasoning will have relatively more explanatory power in databases of rental apartments, whereas rule-based reasoning will have a relative advantage in sales data. We motivate this hypothesis on theoretical grounds, and find empirical support for it by comparing the two statistical techniques (rule-based and case-based) on two databases (rentals and sales).
An agent is asked to assess a real-valued variable Yp based on certain characteristics Xp = (X1p,…,Xmp), and on a database consisting (X1i,…,Xmi,Yi) for i = 1,…,n. A possible approach to combine past observations of X and Y with the current values of X to generate an assessment of Y is similarity-weighted averaging. It suggests that the predicted value of Y, Ysp, be the weighted average of all previously observed values Yi, where the weight of Yi, for every i =1,…,n, is the similarity between the vector X1p,…,Xmp, associated with Yp, and the previously observed vector, X1i,…,Xmi. We axiomatize this rule. We assume that, given every database, a predictor has a ranking over possible values, and we show that certain reasonable conditions on these rankings imply that they are determined by the proximity to a similarity-weighted average for a certain similarity function. The axiomatization does not suggest a particular similarity function, or even a particular functional form of this function. We therefore proceed to suggest that the similarity function be estimated from past observations. We develop tools of statistical inference for parametric estimation of the similarity function, for the case of a continuous as well as a discrete variable. Finally, we discuss the relationship of the proposed method to other methods of estimation and prediction.
This paper determines coverage probability errors of both delta method and parametric bootstrap confidence intervals (CIs) for the covariance parameters of stationary long-memory Gaussian time series. CIs for the long-memory parameter d0 are included. The results establish that the bootstrap provides higher-order improvements over the delta method. Analogous results are given for tests. The CIs and tests are based on one or other of two approximate maximum likelihood estimators. The first estimator solves the first-order conditions with respect to the covariance parameters of a “plug-in” log-likelihood function that has the unknown mean replaced by the sample mean. The second estimator does likewise for a plug-in Whittle log-likelihood.
The magnitudes of the coverage probability errors for one-sided bootstrap CIs for covariance parameters for long-memory time series are shown to be essentially the same as they are with iid data. This occurs even though the mean of the time series cannot be estimated at the usual n1/2 rate.
Keywords: Asymptotics, confidence intervals, delta method, Edgeworth expansion, Gaussian process, long memory, maximum likelihood estimator, parametric bootstrap, t statistic, Whittle likelihood
In this paper, we prove the validity of an Edgeworth expansion to the distribution of the Whittle maximum likelihood estimator for stationary long-memory Gaussian models with unknown parameter . The error of the (s-2)-order expansion is shown to be o(n(s-2)/2) – the usual iid rate — for a wide range of models, including the popular ARFIMA(p,d,q) models. The expansion is valid under mild assumptions on the behavior of spectral density and its derivatives in the neighborhood of the origin. As a by-product, we generalize a Theorem by Fox and Taqqu (1987) concerning the asymptotic behavior of Toeplitz matrices.
Lieberman, Rousseau, and Zucker (2002) (LRZ) establish a valid Edgeworth expansion for the maximum likelihood estimator for stationary long-memory Gaussian models. For a significant class of models, their expansion is shown to have an error of o(n-1). The results given here improve upon those of LRZ in that the results provide an Edgeworth expansion for an asymptotically efficient estimator, as LRZ do, but the error of the expansion is shown to be o(n-(s-2)/2), not o(n-1), for a broad range of models.
Keywords: ARFIMA, Edgeworth expansion, Long Memory, Whittle estimator
We provide in this paper asymptotic theory for the multivariate GARCH (p,q) process. Strong consistency of the quasi-maximum likelihood estimator (MLE) is established by appealing to conditions given in Jeantheau [19] in conjunction with a result given by Boussama [9] concerning the existence of a stationary and ergodic solution to the multivariate GARCH (p,q) process. We prove asymptotic normality of the quasi-MLE when the initial state is either stationary or fixed.
. The error of the (s-2)-order expansion is shown to be o(n(s-2)/2) – the usual iid rate — for a wide range of models, including the popular ARFIMA(p,d,q) models. The expansion is valid under mild assumptions on the behavior of spectral density and its derivatives in the neighborhood of the origin. As a by-product, we generalize a Theorem by Fox and Taqqu (1987) concerning the asymptotic behavior of Toeplitz matrices.